where \(\nabla \)  denotes the gradient operator. Write \(\alpha _N=N\omega _{N-1}^{1/(N-1)}\) , where \(\omega _{N-1}\)  stands for
the area of the unit sphere in \(\mathbb {R}^N\) . Then the classical Trudinger-Moser inequality [1]}, [2]}, [3]}, [4]}, [5]} says
\(\sup _{u\in W_0^{1,N}(\Omega ),\,\Vert u\Vert _{W_0^{1,N}(\Omega )}\le 1}\int _\Omega e^{\alpha |u|^{\frac{N}{N-1}}}dx<\infty ,\quad \forall \alpha \le \alpha _N.\) 
